First and foremost, if there's ever a problem try to perform pin hole reset

Bluetooth (2.0+EDR required)
Bluetooth dongles (especially built-in ones) may be fussy with drivers (device manager is your best friend)
Remember, you might then need to disable automatic driver updating to prevent windows from overriding them

If yours's not supported zadig will help.
If it suddenly stop to respond (or acts weirdly) try to maintain line of sight and avoid USB 1.x ports

If nothing happen eventually, Windows Event Log service should be to blame (or Network List Service)

EDIT: and if nothing still happens... well, I dunno. After a reboot, laptop BT built-in adapter (which I even had to force with zadig because asus has a nice way to rebrand qualcomm stuff and change VID) didn't seem to work anymore. ~~Nefarious fork worked though.~~ FIXED

Controllers
Of course you can use the controllers that came with your Playstation (screw you microsoft accessories mob)
Aside of them though, there are no precise guarantee.
Namely: blatantly crappy chinese replacements selling for as low as 12€ worked and all, while fakes done so well you'd need a magnyfing glass to spot differences, didn't work wirelessly [possible workaround]
How to detect: visual check, led behavior, enumeration constants, ~~MAC address~~THIS, interiors

Topic on Controller talk:DualShock 3